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Interface Conductance Modal Analysis (ICMA)

The interface conductance modal analysis (ICMA) method is a formalism we developed to enable the study of phonon transport at interfaces. In essence it is based on the same underlying principle as GKMA, but here it is reformulated for an interface. One of the most interesting things about ICMA is that it treats the problem of phonon transport at an interface in terms of correlation rather than scattering. This is distinctly different from the predominant physical picture that has been used for many decades. While developing ICMA we realized that you have to perform a full super cell lattice dynamics calculation of a combined structure containing both materials to find the right normal modes needed to describe what happens at the interface. What was surprising though, was that different types of modes exist, namely extended modes, partially extended modes, interfacial modes and isolated modes. These new classifications have prompted a rethinking of the more widely accepted physical picture, as our papers have shown a number of non-intuitive behaviors are possible within our new ICMA based paradigm.

The interface conductance modal analysis (ICMA) method works by first calculating the net heat flow across an interface at a given instant during a molecular dynamics simulation. The heat flow is determined from the product of the forces between atoms on the two sides of the interface and their respective velocities. This net heat flow is then tracked over time and the degree to which it remains correlated in time is proportional to the interface conductance between the two groups of atoms. The key advancement of ICMA, however, is in the fact that it provides the normal mode level details of such a calculation, by enabling us to calculate each individual mode’s contribution to the interface conductance.

In order to calculate the normal mode contributions, we must first determine the modes from a lattice dynamics calculation. When we initially developed ICMA, we tried using the modes that had previously been envisioned as the correct basis set, namely if one applies the phonon gas model (PGM), and uses expressions derived from detailed balance, they suggest that one should only need information about the modes from one side of the interface, either material A or material B. However, the expression for conductance in ICMA suggests 3 choices might be valid. Specifically, using the modes of material A alone, denoted by {A}, adding the modes from material A and B together, denoted by {A+B}, or using the modes that would be obtained from a lattice dynamics calculation for the entire structure of the two materials joined, denoted by {AB}. The question is then, which set of modes is the correct choice?

To determine which set of modes is correct, we devised a test, whereby we launch a wave packet (WP) towards an interface, and we note 4 requirements that must be strictly satisfied for a modal basis set to be valid. First, as the WP approaches the net heat flow across the interface should be zero, until the WP arrives at the interface. Second, the net heat flow summed up for all the modes should integrate to the total amount of energy transmitted across the interface. Third, since this is a carefully designed WP that experiences a purely elastic scattering event, a correct basis set should show no excitation of frequencies outside the original band of frequencies used to create the WP – as a Fourier Transform of the MD simulation shows that only the originally excited frequencies ever get excited. Lastly, since no outside frequencies should get excited, there should also be no net heat flow contributions for modes outside the originally excited band of frequencies.

When we did the WP test, we found that all 3 choices of modal basis set met requirements 1 and 2, but only the choice {AB} met requirements 3 and 4. As the picture shows, before an after the WP impact all basis set choices exhibit the correct behavior. However, during impact, only {AB} shows the correct frequency range. Thus, it was concluded that only choice {AB} is the correct basis set.

The fact that only the choice {AB} yields the correct behavior has important implications, because the {AB} basis set contains modes that differ from the conventional physical picture described by the phonon gas model (PGM). Generally, the {AB} basis set contains extended modes, that are shared by both materials, partially extended modes that are primarily in one material, but partly extend into the other material, isolated modes, which are confined to one material, and interfacial modes, which are localized at the interface. What is most interesting about these mode classifications are the interfacial modes, which are localized and therefore cannot impinge on the interface as is described by the PGM physical picture. As a result, it raises questions around the validity of the PGM physical picture, since localized interfacial modes fall outside that paradigm.

One important question that developed immediately after discovering the existence of interfacial modes was whether or not they’re negligible. One easy way to reconcile application of the PGM is if it turned out that interfacial modes could actually be neglected, which would then allow the PGM to still prevail as a valid physical picture, from an engineering perspective. However, we studied a fictitious system of silicon in contact with a silicon that was artificially given a 4X larger mass. This creates different vibrational frequencies in the second (heavier) material, and produces interfacial resistance. The accumulation is shown in the figure (top) and showed a very large jump between 12-13 THz. Interestingly, the density of states (DoS) did not show there were very many modes in this frequency range (see bottom of the figure). We then realized that this very strong set of contributions comes from a very small group of interfacial modes, which nullified the hypothesis that interfacial modes can in general be treated as negligible.

We later found that interfacial modes are not only non-negligible, but they have the largest contributions to the conductance on a per mode basis. Another way of redeeming the PGM, however, is to consider that the interfacial modes can somehow still be treated within the PGM framework, e.g., by assigning them a velocity and a transmission probability of unity. However, the figure (top) shows what the accumulation would look like if we assigned the highest possible velocity (i.e., the speed of sound) to all modes and gave every mode a transmission probability of unity. The slope of this curve then provides the maximum conductance contribution that any mode can have, and the figure (bottom) shows that where the large contributions from the interfacial modes occurs, actually exceeds the PGM. This is to say that ICMA predicts conductance contributions for the interfacial modes that are 7X higher than the highest contribution that can ever be rationalized by the PGM. These results accompanied by others that followed all point towards the need to develop a new physical picture that goes beyond the PGM to describe what happens at interfaces, which is now what our work is focused on.